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Number of solutions to a system of equations graphically

Sal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16. Created by Sal Khan.

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Video transcript

So that we don't get stumped by the Arbeglas in our life and especially when we don't have talking birds to help us, we should be able to identify when things get a little bit weird with our systems of equations. When we have scenarios that have an infinite number of solutions or that have no solutions at all. And just as a little bit of a review of what could happen, these are the-- think about the three scenarios. You have the first scenario which is kind of where we started off, where you have two systems that just intersect in one place. And then you have essentially one solution. So if you were to graphically represent it you have one solution right over there, one solution. And this means that the two constraints are consistent and the two constraints are independent of each other. They're not the exact same line, consistent and independent. Then you have the other scenario where they're consistent, they intersect,but they're essentially the same line. They intersect everywhere. So this is one of the constraints for one of the equations, and the other one if you look at it, if you graph it, it is actually the exact same one. So here you have an infinite number of solutions. It's consistent, you do have solutions here, but they're dependent equations. It's a dependent system. And then the last scenario, and this is when you're dealing in two dimensions, the last scenario is where your two constraints just don't intersect with each other. One might look like this, and then the other might look like this. They have the exact same slope but they have different intercepts. So this there is no solution, they never intersect. And we call this an inconsistent system. And if you wanted to think about what would happen just think about what's going on here. Here you have different slopes. And if you think about it, two different lines with different slopes are definitely going to intersect in exactly one place. Here they have the same slope and same y-intercept, so you have an infinite number of solutions. Here you have the same slope but different y-intercepts, and you get no solutions. So the times when you're solving systems where things are going to get a little bit weird are when you have the same slope. And if you think about it, what defines the slope, and I encourage you to test this out with different equations, is when you have-- if you have your x's and y's, or you have your a's and b's or you have your variables on the same side of an equation, where they have the same ratio with respect to each other. So with that, kind of keeping that in mind, let's see if we can think about what types of solutions we might find. So let's take this down. So they say determine how many solutions exist for the system of equations. So you have 10x minus 2y is equal to 4, and 10x minus 2y is equal to 16. So just based on what we just talked about the x's and the y's are on the same side of the equation and the ratio is 10 to negative 2. Same ratio. So something strange is going to happen here. But when we have the same kind of combination of x's and y's in the first one we get 4, and on the second one we get 16. So that seems a little bit bizarre. Another way to think about it, we have the same number of x's, the same number of y's but we got a different number on the right hand side. So if you were to simplify this, and we could even look at the hints to see what it says, you'll see that you're going to end up with the same slope but different y-intercepts. So we convert both the slope intercept form right over here and you see one, the blue one is y is equal to 5x minus 2, and the green one is y is equal to 5x minus 8. Same slope, same ratio between the x's and the Y's, but you have different values right over here. You have different y-intercepts. So here you have no solutions. That is this scenario right over here if you were to graph it. So no solutions, check our answer. Let's go to the next question. So let's look at this one right over here. So we have negative 5 times x and negative 1 times y. We have 4 times x and 1 times y. So it looks like the ratio if then we're looking at the x's and y's always on the left hand side right over here, it looks like the ratios of x's and y's are different. You have essentially 5 x's for every one y, or you could say negative 5 x's for every negative 1 y, and here you have 4 x's for every 1 y. So this is fundamentally a different ratio. So right off the bat you could say well these are going to intersect in exactly one place. If you were to put this into slope intercept form, you will see that they have different slopes. So you could say this has one solution and you can check your answer. And you could look at the solution just to verify. And I encourage you to do this. So you see the blue one if you put in the slope intercept form negative 5x plus 10 and you take the green one into slope intercept form negative 4x minus 8. So different slopes, they're definitely going to intersect in exactly one place. You're going to have one solution. Let's try another one. So here we have 2x plus y is equal to negative 3. And this is pretty clear, you have 2x plus y is equal to negative 3. These are the exact same equations. So it's consistent information, there's definitely solutions. But there's an infinite number of solutions right over here. This is a dependent system. So there are infinite number of solutions here and we can check our answer. Let's do one more because that was a little bit too easy. OK so this is interesting right over here, we have it in different forms. 2x plus y is equal to negative 4, y is equal to negative 2x minus 4. So let's take this first blue equation and put it into slope intercept form. If we did that you would get, if you just subtract 2x from both sides you get y is equal to negative 2x minus 4, which is the exact same thing as this equation right over here. So once again they're the exact same equation. You have an infinite number of solutions. Check our answer, and you can look at the solution right here. You convert the blue one into slope intercept and you get the exact same thing as what you saw in the green one.